A259834 Number of permutations of [n] with no fixed points where the maximal displacement of an element equals n-1.
0, 0, 1, 2, 5, 20, 97, 574, 3973, 31520, 281825, 2803418, 30704101, 367114252, 4757800705, 66432995030, 994204132517, 15875195019224, 269397248811073, 4841453414347570, 91856764780324165, 1834779993945449348, 38485629141294791201, 845788826477292504302
Offset: 0
Examples
a(2) = 1: 21. a(3) = 2: 231, 312. a(4) = 5: 2341, 3421, 4123, 4312, 4321.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..450
Programs
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Maple
a:= proc(n) option remember; `if`(n<3, [0, 0, 1][n+1], ((2*n^2-11*n+13)*a(n-1) +(2*n-5)*(n-3)*a(n-2))/(2*n-7)) end: seq(a(n), n=0..30); -
Mathematica
Join[{0, 0}, Table[DifferenceRoot[Function[{y, m}, {y[1 + m] == (n - m)*y[m] + (n - m) y[m - 1], y[0] == 0, y[1] == 1, y[2] == 1}]][n], {n, 2, 30}]] (* Benedict W. J. Irwin, Nov 03 2016 *) Table[If[n<2, 0, Subfactorial[n-2]+2*Subfactorial[n-1]], {n,0,23}] (* Peter Luschny, Oct 04 2017 *) -
Python
def A259834_list(len): L, u, x, y = [0], 1, 0, 0 for n in range(len): y, x, u = x, x*n + u, -u L.append(y + 2*x) L[1] = 0 return L print(A259834_list(23)) # Peter Luschny, Oct 04 2017
Formula
a(n) = ((2*n^2-11*n+13)*a(n-1) + (2*n-5)*(n-3)*a(n-2))/(2*n-7) for n > 2.
a(n) = (n-2)! * [x^(n-2)] exp(-x)*(x+1)/(x-1)^2 for n > 1.
a(n) ~ 2 * (n-1)! / exp(1). - Vaclav Kotesovec, Jul 07 2015
a(n) = y(n,n), n > 1, where y(m+1,n) = (n-m)*y(m,n) + (n-m)*y(m-1,n), with y(0,n)=0, y(1,n)=y(2,n)=1 for all n. - Benedict W. J. Irwin, Nov 03 2016
From Peter Luschny, Oct 05 2017: (Start)
a(n) = (Gamma(n-1, -1) + 2*Gamma(n, -1)) / e for n >= 2.
a(n) = (2*n-1)*A000166(n-2) - 2*(-1)^n for n >= 2.
a(n+2) = (-1)^n*(-hypergeom([1,1-n], [], 1) + hypergeom([2,2-n], [], 1)) = A292898(2, n) for n >= 0.
a(n) ~ 2*sqrt(2*Pi)*exp(-n-1)*n^(n-1/2). (End)
a(n+2) = A306455(n) + n! for n >= 0. - Alois P. Heinz, Feb 16 2019
Comments